solve the following systwm graphically. Be sure to check your solution. if the system has an infinite number of solutions, use set-builder notation to write no solution, state this. 4y=x+8, x=2/3y+2

http://www.wolframalpha.com/input/?i=plot+4y%3Dx%2B8%2C+x%3D2%2F3y%2B2

To solve the given system of equations graphically, we will plot the graphs of both equations on the same coordinate plane and find the point of intersection, if any. Let's start by rewriting each equation in slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept.

Equation 1: 4y = x + 8
Dividing both sides of the equation by 4, we get:
y = (1/4)x + 2

Equation 2: x = (2/3)y + 2
Rewriting the equation by switching the positions of x and y, we get:
y = (3/2)(x - 2)

Now, we can plot the graphs of these equations on the same coordinate plane:

1. Plot the graph of y = (1/4)x + 2:
- Start by plotting the y-intercept, which is the point (0, 2).
- Then use the slope, which is 1/4, to find additional points. For example, if you move one unit to the right (x+1) and four units up (y+4), you will get the point (1, 6). You can continue this process to plot more points if needed.

2. Plot the graph of y = (3/2)(x - 2):
- Start by plotting the point (2,0), which is the x-intercept.
- Then use the slope, which is 3/2, to find additional points. For example, if you move two units to the right (x+2) and three units up (y+3), you will get the point (4, 3). Again, you can plot more points if desired.

Once you have both graphs plotted, check for their point of intersection. If the two lines intersect at a single point, that point represents the solution to the system of equations.

Finally, to double-check your solution, substitute the x and y values of the intersection point into both original equations to ensure they satisfy both equations.

If the two lines are parallel and never intersect, the system has no solution and you should state this using set-builder notation.